28 problems

GMPS representation conjecture. All pure $N$-mode Gaussian states can be described as GMPS.

A finitely correlated state is a pure state on a quantum spin chain with an associated finite-dimensional memory system. Consider a spin and a subset of other spins, and suppose th…

The MPS construction gives an upper bound on the number of non-zero entanglement eigenvalues of the ground state. Jones's optimality conjecture. For $N=2$, this construction should…

A fuzzy sphere is a noncommutative geometric configuration represented by matrices, and an integrable matrix product state (MPS) is a matrix product state satisfying the integrabil…

Let $d(n)$ denote the minimum bond dimension of a translation-invariant matrix product state representation with periodic boundary conditions for the $W$-state of order $n$. Garcia…

Let $d(\psi)$ denote the minimum bond dimension of a translation-invariant matrix product state (TI MPS) representation with periodic boundary conditions (PBC) of a state $\psi$, a…

Let $\mathscr{Q}_{MPS}$ denote the proposed space of states represented by normal matrix product state tensors of finite bond dimension, namely … A parametrized phase in spatial di…

Let $S=\{A_1,\ldots,A_g\}\subseteq M_n(\mathbb{C})$ satisfy condition $C1$, meaning that words of some common length span $M_n(\mathbb{C})$, and let $\Gamma_L$ be the associated ma…

For $S=\{A_1,\ldots,A_g\}\subseteq M_n(\mathbb{C})$, define … The condition that words of length $L$ span $M_n(\mathbb{C})$ is equivalent to injectivity of $\Gamma_L$. Pérez-García…

Let $M,N\in \operatorname{Mat}_{D\times D}^d$, and write $\lvert M_n\rangle$ and $\lvert N_n\rangle$ for the corresponding matrix product states of system size $n$. The tensors are…

Gaussian MPS convergence conjecture. As the bond dimension $q$ increases, the normalized injective norm of the Gaussian MPS converges to that of the corresponding Gaussian tensor:

Let $\chi$ be the bond dimension, and let $n_z$ and $n_Z$ denote the numbers of the corresponding parameters in the model. Assume the generic condition that … are pairwise distinct…

Let $\rho$ be a matrix product density operator (MPDO), and let $I_\rho(A:C\mid B)$ denote its conditional mutual information for a tripartition into regions $A$, $B$, and $C$: … w…

Positivity and definiteness conjecture. The dominant eigenvalue $\eta$ is real and positive, and

Generic MPDO conjecture. For a generic' MPDO, there exists a constant $c>0$ such that for any tripartition $ABC$ in which $B$ separates $A$ from $C$ by distance $\ell$,

Let the target functions be represented on systems whose size increases, and let an MPS representation have bond dimension $\chi=2$. Constant-bond-dimension MPS approximation conje…

Closedness conjecture. Except for the trivial cases in Proposition, $\operatorname{uMPS}(D,d,N)$ is only closed if it fills the ambient space $\operatorname{Cyc}^N(\mathbb{C}^d)$.

Generic injectivity index conjecture. The generic injectivity index satisfies

Let $L$ be a linear space of $D\times D$ matrices, and let $L^k$ denote the linear space spanned by all products of $k$ matrices in $L$. If … for some $k$, then … for all…

The states $|\chi_0^{(k)}\rangle$ arise as zero-momentum components of periodic matrix product states. For the cases explicitly obtained, they can be expressed using products invol…

In one dimension, consider eigenvectors of a many-body-localised Hamiltonian. A matrix-product state vector has the form … where $D$ is the bond dimension and…

Let $\phi_N:\mathbb{C}^5\to\overline{\operatorname{PB}}(2,2,N)$ be the trace parameterization of periodic boundary matrix product states arising from the trace algebra for two…

Let $\Psi$ be a translation-invariant $N$-qubit matrix product state with open boundary conditions and bond and physical dimensions $D=d=2$. A reduced density operator on a chain o…

Let $N\geq 4$. An improper marginalization maps a translation-invariant $N$-qubit matrix product state variety $\overline{\operatorname{OB}}(2,2,N)$ to…

Spectral conjecture. For general $S$, the eigenvalues of $G$ are